Isoperimetric profiles and random walks on some groups defined by piecewise actions

TL;DR

This paper investigates the isoperimetric and spectral properties of specific finitely generated groups constructed through actions on Schreier graphs and gluing techniques, revealing new estimates and raising open questions.

Contribution

It introduces a novel construction called pocket-extension and analyzes the isoperimetric and spectral profiles of these groups, providing new bounds and insights.

Findings

Obtained sharp estimates for isoperimetric profiles.

Analyzed spectral profiles of groups defined by piecewise actions.

Identified open questions for further research.

Abstract

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em pocket-extension} of a group $G$---this leads to the study of certain finitely generated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$. Some sharp estimates are obtained while many challenging questions remain.